## Abstract

Consider k independent random samples from p-dimensional multivariate normal distributions. We are interested in the limiting distribution of the log-likelihood ratio test statistics for testing for the equality of k covariance matrices. It is well known from classical multivariate statistics that the limit is a chi-square distribution when k and p are fixed integers. Jiang and Qi (Scand J Stat 42:988–1009, 2015) and Jiang and Yang (Ann Stat 41(4):2029–2074, 2013) have obtained the central limit theorem for the log-likelihood ratio test statistics when the dimensionality p goes to infinity with the sample sizes. In this paper, we derive the central limit theorem when either p or k goes to infinity. We also propose adjusted test statistics which can be well approximated by chi-squared distributions regardless of values for p and k. Furthermore, we present numerical simulation results to evaluate the performance of our adjusted test statistics and the log-likelihood ratio statistics based on classical chi-square approximation and the normal approximation.

Original language | English (US) |
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Pages (from-to) | 247-279 |

Number of pages | 33 |

Journal | Metrika |

Volume | 87 |

Issue number | 3 |

State | Published - Apr 2024 |

### Bibliographical note

Publisher Copyright:© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023.

## Keywords

- Central limit theorem
- Likelihood ratio test
- Multivariate gamma function
- Multivariate normal distribution